VNU Journal of Science, Mathematics - Physics 26 (2010) 147-154
147
Calculation of Lindemann’s melting Temperature
and Eutectic Point of bcc Binary Alloys
Nguyen Van Hung
*
, Nguyen Cong Toan, Hoang Thi Khanh Giang

Department of Physics, University of Science, VNU
334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
Received 1 June 2010
Abstract. Analytical expressions for the ratio of the root mean square fluctuation in atomic
positions on the equilibrium lattice positions and the nearest neighbor distance and the mean
melting curves of bcc binary alloys have been derived. This melting curve provides information on
Lindemann’s melting temperatures of binary alloys with respect to any proportion of constituent
elements and on their euctectic points. Numerical results for some bcc binary alloys are found to
be in agreement with experiment.
Keywords: Lindemann’s melting temperature, eutectic point, bcc binary alloys.
1. Introduction
The melting of materials has great scientific and technological interest. The problem is to
understand how to determine the temperature at which a solid melts, i.e., its melting temperature. The
atomic vibrational theory has been successfully applied by Lindemann and others [1-5]. The
Lindemann’s criterion [1] is based on the concept that the melting occurs when the ratio of the root
mean square fluctuation (RMSF) in atomic positions on the equilibrium lattice positions and the
nearest neighbor distance reaches a critical value. Hence, the lattice thermodynamic theory is one of
the most important fundamentals for interpreting thermodynamic properties and melting of materials
[1-6, 8-15]. The binary alloys have phase diagrams containing the liquidus or melting curve going
from the point corresponding the melting temperature of the host element to the one of the doping
element. The minimum of this melting curve is called the eutectic point. The melting is studied by
experiment [7] and by different theoretical methods. X-ray Absorption Fine Structure (XAFS)
procedure in studying melting [8] is focused mainly on the Fourier transform magnitudes and

(
)
∑∑
−−
+=+=
q
n
i
q
n
i
q
q
n
n
i
q
n
i
qn
eeee
q.Rq.Rq.Rq.R
uuUuuU
*
222
*
111
2
1
,

W uK. , (3)
where K is the scattering vector equaling a reciprocal lattice vector, and
q
u is the mean atomic
vibration amplitude.
It is apparent that 1/8 atom on the vertex and one atom in the center of the bcc are localized in an
elementary cell. Hence, the total number of atoms in an elementary cell is 2. Then if on average s is
atomic number of type 1 and (2 - s) is atomic number of type 2, the quantity
q
u is given by

(
)
2
2
21 qq
q
ss uu
u
−
+
= . (4)
The potential energy of an oscillator is equal to its kinetic energy so that the mean energy of atom
k vibrating with wave vector q has the form

2
kqkq
uM
&
=ε . (5)

2
and
q
u
1
[13], i.e.,

2112
/, MMmmuu
qq
== , (7)
and Eqs. (5, 6) we obtain the mean energy for the atomic vibration with wave vector q
N.V. Hung et al. / VNU Journal of Science, Mathematics - Physics 26 (2010) 147-154
149

( )
[
]
2
211
2
2
2
msMsMuN
qqq
−+= ωε . (8)
The mean energy for this qth lattice mode calculated using the phonon energy with
q
n as the mean
number of oscillators is given by

=
2
2
1
2
1
2
1
h
. (10)
Using Eq. (4) and Eq. (7) the mean atomic vibration amplitude has the form

 
22
2
1
1
2
4
qq
ussmu



. (11)
To study the MSD Eq. (3) we use the Debye model, where all three vibrations have the same
velocity [3]. Hence, for each polarization with taking Eq. (11) into account we get the mean value

( )
[ ]






+
−+==
∑∑
2
2
1
)2(
4
1
2
1
1
2
2
2
2
ω
h
. (13)
Transforming the sum over q into the corresponding integral [3], Eq. (13) is changed into the
following form

[ ]
ω
ωω

1
1
)2(
4
1
/
h
h
h
. (14)
Denoting
DDBB
kTkz ωθω hh == ,/ with
DD
θω , as Debye frequency and temperature,
respectively, we obtain

[ ]
dzz
ekM
T
mssKW
T
z
DB
D
∫




z
e
z
, and 0
2
→
z
, then the DWF Eq. (15) with using Eq. (7) is given by

[
]
2
21
22
12
)2(
4
3
DB
kMM
TKMssM
W
θ
h−+
=
, (16)
which is linearly proportional to the temperature T as it was shown already [3, 14].
From Eq. (12) with using Eq. (3) for W we obtain
N.V. Hung et al. / VNU Journal of Science, Mathematics - Physics 26 (2010) 147-154
150

&
E
. (18)
Using this expression and Eqs. (6, 7) we obtain the atomic MSF in the form

∑∑
=
q
q
n
n
umU
N
2
1
2
2
2
1
, (19)
which by using Eq. (17) is given by

[]
2
2
2
2
2
)2(
24

=
∑
h
. (21)
Hence, at
D
T θ>> the MSF in atomic positions about the equilibrium lattice positions is
determined by Eq. (21) which is linearly proportional to the temperature T.
Therefore, at a given temperature T the quantity R defined by the ratio of the RMSF in
atomic positions about the equilibrium lattice positions and the nearest neighbor distance d is
given by

[ ]
22
1
22
)2(
18
dkmssM
Tm
R
DB
θ−+
=
h
. (22)
Based on the Lindemann’s criterion the binary alloy will be melted when this value R
reaches a threshold value R
m
, then the Lindemann’s melting temperature

222
1
,
h
θ
χ . (23)
If we denote x as proportion of the mass of the element 1 in the binary alloy, then we have

( )
21
1
2 MssM
sM
x
−+
= . (24)
From this equation we obtain the mean number of atoms in the element 1 for each binary alloy
lattice cell

xxm
x
s
+−
=
)1(
2
. (25)
We consider one element to be the host and another dopant. If the tendency to be the host is equal
for both constituent elements, we can take averaging the parameter m with respect to the atomic mass
proportion of the constituent elements in alloy as follows

( ) ( )
011
1
2
2
1
2
=−






−−+−
M
M
xm
M
M
xxmx
, (27)
which provides the following solution

( )
( )
( ) ( )
1
2
2



−−−
= , (28)
replacing m in Eq. (23) for the calculation of Lindemann’s melting temperatures.
The threshold value R
m
of the ratio of RMSF in atomic positions on the equilibrium lattice
positions and the nearest neighbor distance at the melting is contained in
χ
which will be obtained by
an averaging procedure. The average of
χ
can not be directly based on
1
χ and
2
χ because it has the
form of Eq. (23) containing
2
m
R , i.e., the second order of
m
R , while the other averages have been
realized based on the first order of the displacement as Eq. (22). That is why we have to perform
average for
2/1
χ and then obtain

( )

different pairs of elements with the masses M
1
and M
2
of the same bcc structure.
The eutectic point is calculated using the condition for minimum of the melting curve, i.e.,
0=
dx
dT
m
. (31)

3. Numerical results and comparison to experiment
Now we apply the derived theory to numerical calculations for bcc binary alloys. According to the
phenomenological theory (PT) [10] Figure 1 shows the typical possible phase diagrams of a binary
alloy formed by the components A and B, i.e., the dependence of temperature T on the proportion x of
N.V. Hung et al. / VNU Journal of Science, Mathematics - Physics 26 (2010) 147-154
152

0 0.2 0.4 0.6 0.8 1
280
285
290
295
300
305
310
315
320
325

Eutectic point, Expt., Ref. 7
Cr
1-x
Mo
x
element B doped in the host element A. Below isotropic liquid mixture L, the liquidus or melting
curve beginning from the melting temperature T
A
of the host element A passes through a temperature
minimum T
E
known as the eutectic point E and ends at the melting temperature T
B
of the doping
element B. The phase diagrams contain two solid crystalline phases α and β. The eutectic point is
varied along the eutectic isotherm T = T
E
. The eutectic temperature T
E
can be a value lower T
A
and T
B

(Figure 1a) or in the limiting cases equaling T
A
(Figure 1b) or T
B
(Figure 1c). The mass proportion x
characterizes actually the proportion of doping element mixed in the host element to form binary alloy.

= 288 K and the
eutectic proportion x
E
= 0.3212 are in a reasonable agreement with the experimental values T
E
= 285.8
K and x
E
= 0.35 [7], respectively. For Cr
1-x
Mo
x
the calculated eutectic temperature T
E
= 2125 K agrees
N.V. Hung et al. / VNU Journal of Science, Mathematics - Physics 26 (2010) 147-154
153
0 0.2 0.4 0.6 0.8 1
1800
1850
1900
1950
2000
2050
2100
2150
2200
2250
2300
Mass proportion x of V

a reasonable agreement with the experimental value x
E
= 0.20 [7]. Figure 3 shows that our calculated
melting curve for Fe
1-x
V
x
corresponds to the phase diagram of Figure 1b and for Cr
1-x
Cs
x
to those of
Figure 1c of the PT. Table 1 shows the good agreement of the Lindemann’s melting temperatures
taken from the calculated melting curve with respect to different proportions of constituent elements of
binary alloy Cs
1-x
Rb
x
with experimental values [7].
4. Conclusions
In this work a lattice thermodynamic theory on the melting curves, eutectic points and eutectic
isotherms of bcc binary alloys has been derived. Our development is derivation of analytical
expressions for the melting curves providing information on Lindemann’smelting temperatures with
respect to different proportions of constituent elements and eutectic points of the binary alloys.
The significance of the derived theory is that the calculated melting curves of binary alloys
correspond to the experimental phase diagrams and to those qualitatively shown by the
phenomenological theory. The Lindemann’s melting temperatures of a considered binary alloy change
from the melting temperature of the host element when the whole elementary cell is occupied by the
atoms of the host element to those of binary alloy with respect to different increasing proportions of
the doping element and end at the one of the pure doping element when the whole elementary cell is
occupied by the atoms of the doping element.

N.V. Hung et al. / VNU Journal of Science, Mathematics - Physics 26 (2010) 147-154
154

Acknowledgments. This work is supported by the research project QG.08.02 and by the research
project No. 103.01.09.09 of NAFOSTED.
References
[1] F.A. Lindemann, Z. Phys. 11 (1910) 609.
[2] N. Snapipiro, Phys. Rev. B 1 (1970) 3982.
[3] J.M. Ziman, Principles of the Theory of Solids, Cambrige University Press, London, 1972.
[4] H.H. Wolf, R. Jeanloz, J. Geophys. Res. 89 (1984) 7821.
[5] R.K. Gupta, Indian J. Phys. A 59 (1985) 315.
[6] Charles Kittel, Introduction to Solid State Physics, 3rd Edition (Wiley, New York, 1986).
[7] T.B. Massalski, Binary Alloy Phase Diagrams, 2
nd
ed. (ASM Intern. Materials Parks, OH, 1990).
[8] E.A. Stern, P. Livins, Zhe Zhang, Phys. Rev B, Vol. 43, No.11 (1991) 8850.
[9] D. Alfè, L. Vočadlo, G.D. Price, M.J. Gillan, J. Phys.: Condens. Matter 16 (2004) S937.